R. M. A. Azzam and Bruce E. Perilloux, "Constraint on the optical constants of a film–substrate system for operation as an external-reflection retarder at a given angle of incidence," Appl. Opt. 24, 1171-1179 (1985)

Abstract

Given a transparent film of refractive index n1 on an absorbing substrate of complex refractive index n2-jk2, we examine the constraint on n1, n2, and k2 such that the film–substrate system acts as an external-reflection retarder of specified retardance Δ at a specified angle of incidence ϕ. The constraint, which takes the form f(n1,n2,k2;ϕ,Δ) = 0, is portrayed graphically by equi-n1 contours in the n2,k2 plane at ϕ = 45, 70° and for Δ = ±90 and ±180°, corresponding to quarterwave and halfwave retarders (QWR and HWR), respectively. The required film thickness as a fraction of the film thickness period and the polarization-independent device reflectance ℛ are also studied graphically as functions of the optical constants. It is found that as n2 → 0, ℛ → 1, so that a metal substrate such as Ag is best suited for high-reflectance QWR (ϕ > 45°) and HWR (ϕ ≤ 45°). However, films that achieve QWR at ϕ ≤ 45° are excellent antireflection coatings of the underlying dielectric, semiconductor, or metallic substrate.

If the incident light is linearly polarized at 45° azimuth with respect to the plane of incidence, so that the p and s components of the incident electric vector are equal and inphase, the p component of the electric vector of the reflected light will lead the s component by 90°, when ρ = j, hence the identification of the p direction as the fast axis. The exp(jωt) harmonic time dependence is assumed.

Other (4)

If the incident light is linearly polarized at 45° azimuth with respect to the plane of incidence, so that the p and s components of the incident electric vector are equal and inphase, the p component of the electric vector of the reflected light will lead the s component by 90°, when ρ = j, hence the identification of the p direction as the fast axis. The exp(jωt) harmonic time dependence is assumed.

Figures (18)

Contours of constant film refractive index, n1 = constant, in the n2,k2 plane, where n2-jk2 is the substrate complex refractive index. These equi-n1 contours represent the constraint on the optical constants such that the ratio of the complex p- and s-reflection coefficients of the film–substrate system ρ = j at ϕ = 70° angle of incidence, i.e., a quarterwave retarder (QWR) with p fast axis is realized.

Film thickness as a fraction of the film thickness period ζ, required to achieve ρ = j QWR at ϕ = 70°, plotted as a function of n2 at constant n1 (marked by each curve). n1,n2 specify a particular film–substrate QWR completely, because k2 can be deduced from Fig. 1. The contour AB describes the variation of ζ with n2 along the similarly marked n1 = 1.6 contour in Fig. 1.

Metrics

Table I

Characteristics of Quarterwave Retarders (QWR, ρ = +j) at 70° Angle of Incidence Using a Dielectric Thin Film on a Ag Substrate at Several Wavelengths a

λ(nm)

n2

k2

n1

ζ

d(nm)

ℛ(%)

400

0.075

1.93

1.219154

0.277011

71.33

94.09

500

0.050

2.87

1.291322

0.278893

78.72

97.72

600

0.060

3.75

1.349785

0.285499

88.39

98.20

700

0.075

4.62

1.396728

0.292587

99.10

98.40

800

0.090

5.45

1.433078

0.298824

110.48

98.55

950

0.110

6.56

1.471883

0.306027

128.32

98.72

2000

0.480

14.40

1.601287

0.332817

256.69

98.69

a λ is the wavelength of light. n2,k2 are the real and imaginary parts of the Ag substrate complex refractive index (from Ref. 9). n1 is the film refractive index obtained by solving Eq. (11) with ρ = +j and ϕ = 70°. ζ and d are the normalized and actual (least) film thicknesses, respectively, and ℛ is the polarization-independent reflectance of the QWR.

Table II

Absolute Values of the Magnitude Error (|ρ| − 1) and Phase Error (argρ −90°) Caused by Introducing, One at a Time, an Angle-of-lncidence Error Δϕ = 0.1°, a Film-Refractive-Index Error Δn1 = 0.01, or a Film-Thickness Error Δd = 1 nm to the QWR Designs at ϕ = 70° Listed in Table I

Δϕ = 0.1°

Δn1 = 0.01

Δd = l nm

λ(nm)

Magnitude error

Phase error (deg)

Magnitude error

Phase error (deg)

Magnitude error

Phase error (deg)

400

0.37 × 10−4

0.329

0.198 × 10−2

2.004

0.613 × 10−3

1.246

500

0.61 × 10−5

0.341

0.066 × 10−2

1.044

0.298 × 10−3

0.961

600

0.19 × 10−5

0.344

0.047 × 10−2

0.681

0.245 × l0−3

0.773

700

0.31 × 10−6

0.345

0.038 × l0−2

0.506

0.214 × 10−3

0.645

800

0.49 × 10−6

0.345

0.033 × 10−2

0.414

0.186 × 10−3

0.553

950

0.10 × 10−5

0.345

0.028 × 10−2

0.335

0.151 × 10−3

0.457

2000

0.25 × 10−5

0.345

0.024 × 10−2

0.185

0.093 × 10−3

0.214

Table III

Characteristics of Halfwave Retarders (HWR) at 45° Angle of Incidence Using a Dielectric Thin Film on a Ag Substrate at Several Wavelengths a

λ(nm)

n2

k2

n1

ζ

d(nm)

ℛ(%)

400

0.075

1.93

1.566251

0.417418

59.74

90.89

500

0.050

2.87

1.436138

0.464840

92.97

96.51

600

0.060

3.75

1.385496

0.481921

121.35

97.36

700

0.075

4.62

1.359636

0.489611

147.57

97.76

800

0.090

5.45

1.345145

0.493407

172.49

98.03

950

0.110

6.56

1.333442

0.496091

208.45

98.31

2000

0.480

14.40

1.312227

0.499608

451.99

98.44

a λ is the wavelength of light. n2,k2 are the real and imaginary parts of the Ag substrate complex refractive index (from Ref. 9). n1 is the film refractive index obtained by solving Eq. (11) with ρ = −1 and ϕ = 45°. ζ and d are the normalized and actual (least) film thicknesses, respectively, and ℛ is the polarization-independent reflectance of the HWR.

Tables (4)

Table I

Characteristics of Quarterwave Retarders (QWR, ρ = +j) at 70° Angle of Incidence Using a Dielectric Thin Film on a Ag Substrate at Several Wavelengths a

λ(nm)

n2

k2

n1

ζ

d(nm)

ℛ(%)

400

0.075

1.93

1.219154

0.277011

71.33

94.09

500

0.050

2.87

1.291322

0.278893

78.72

97.72

600

0.060

3.75

1.349785

0.285499

88.39

98.20

700

0.075

4.62

1.396728

0.292587

99.10

98.40

800

0.090

5.45

1.433078

0.298824

110.48

98.55

950

0.110

6.56

1.471883

0.306027

128.32

98.72

2000

0.480

14.40

1.601287

0.332817

256.69

98.69

a λ is the wavelength of light. n2,k2 are the real and imaginary parts of the Ag substrate complex refractive index (from Ref. 9). n1 is the film refractive index obtained by solving Eq. (11) with ρ = +j and ϕ = 70°. ζ and d are the normalized and actual (least) film thicknesses, respectively, and ℛ is the polarization-independent reflectance of the QWR.

Table II

Absolute Values of the Magnitude Error (|ρ| − 1) and Phase Error (argρ −90°) Caused by Introducing, One at a Time, an Angle-of-lncidence Error Δϕ = 0.1°, a Film-Refractive-Index Error Δn1 = 0.01, or a Film-Thickness Error Δd = 1 nm to the QWR Designs at ϕ = 70° Listed in Table I

Δϕ = 0.1°

Δn1 = 0.01

Δd = l nm

λ(nm)

Magnitude error

Phase error (deg)

Magnitude error

Phase error (deg)

Magnitude error

Phase error (deg)

400

0.37 × 10−4

0.329

0.198 × 10−2

2.004

0.613 × 10−3

1.246

500

0.61 × 10−5

0.341

0.066 × 10−2

1.044

0.298 × 10−3

0.961

600

0.19 × 10−5

0.344

0.047 × 10−2

0.681

0.245 × l0−3

0.773

700

0.31 × 10−6

0.345

0.038 × l0−2

0.506

0.214 × 10−3

0.645

800

0.49 × 10−6

0.345

0.033 × 10−2

0.414

0.186 × 10−3

0.553

950

0.10 × 10−5

0.345

0.028 × 10−2

0.335

0.151 × 10−3

0.457

2000

0.25 × 10−5

0.345

0.024 × 10−2

0.185

0.093 × 10−3

0.214

Table III

Characteristics of Halfwave Retarders (HWR) at 45° Angle of Incidence Using a Dielectric Thin Film on a Ag Substrate at Several Wavelengths a

λ(nm)

n2

k2

n1

ζ

d(nm)

ℛ(%)

400

0.075

1.93

1.566251

0.417418

59.74

90.89

500

0.050

2.87

1.436138

0.464840

92.97

96.51

600

0.060

3.75

1.385496

0.481921

121.35

97.36

700

0.075

4.62

1.359636

0.489611

147.57

97.76

800

0.090

5.45

1.345145

0.493407

172.49

98.03

950

0.110

6.56

1.333442

0.496091

208.45

98.31

2000

0.480

14.40

1.312227

0.499608

451.99

98.44

a λ is the wavelength of light. n2,k2 are the real and imaginary parts of the Ag substrate complex refractive index (from Ref. 9). n1 is the film refractive index obtained by solving Eq. (11) with ρ = −1 and ϕ = 45°. ζ and d are the normalized and actual (least) film thicknesses, respectively, and ℛ is the polarization-independent reflectance of the HWR.